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Mirrors > Home > ILE Home > Th. List > exmidontri2or | Unicode version |
Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
Ref | Expression |
---|---|
exmidontri2or |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidontriim 7223 |
. . 3
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2 | onelss 4387 |
. . . . . . . 8
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3 | 2 | adantl 277 |
. . . . . . 7
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4 | orc 712 |
. . . . . . 7
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5 | 3, 4 | syl6 33 |
. . . . . 6
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6 | eqimss 3209 |
. . . . . . . 8
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7 | 6, 4 | syl 14 |
. . . . . . 7
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8 | 7 | a1i 9 |
. . . . . 6
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9 | onelss 4387 |
. . . . . . . 8
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10 | 9 | adantr 276 |
. . . . . . 7
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11 | olc 711 |
. . . . . . 7
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12 | 10, 11 | syl6 33 |
. . . . . 6
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13 | 5, 8, 12 | 3jaod 1304 |
. . . . 5
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14 | 13 | ralimdva 2544 |
. . . 4
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15 | 14 | ralimia 2538 |
. . 3
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16 | 1, 15 | syl 14 |
. 2
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17 | ontri2orexmidim 4571 |
. . . 4
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18 | 17 | adantr 276 |
. . 3
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19 | 18 | exmid1dc 4200 |
. 2
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20 | 16, 19 | impbii 126 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-tr 4102 df-exmid 4195 df-iord 4366 df-on 4368 df-suc 4371 |
This theorem is referenced by: onntri52 7242 |
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